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Chabauty Without the Mordell-Weil Group

  • Michael Stoll
Chapter

Abstract

Based on ideas from recent joint work with Bjorn Poonen, we describe an algorithm that can in certain cases determine the set of rational points on a curve C, given only the p-Selmer group S of its Jacobian (or some other abelian variety C maps to) and the image of the p-Selmer set of C in S. The method is more likely to succeed when the genus is large, which is when it is usually rather difficult to obtain generators of a finite-index subgroup of the Mordell-Weil group, which one would need to apply Chabauty’s method in the usual way. We give some applications, for example to generalized Fermat equations of the form x5 + y5 = z p .

Keywords

Rational points on curves Chabauty’s method Selmer group 

Subject Classification

11G30 14G05 14G25 14H25 11Y50 11D41 

Notes

Acknowledgements

I would like to thank Bjorn Poonen for useful discussions and MIT for its hospitality during a visit of two weeks in May 2015, when these discussions took place. All computations were done using the computer algebra system Magma [3].

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Mathematisches InstitutUniversität BayreuthBayreuthGermany

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